| L(s) = 1 | + 2·3-s + 4-s − 6·7-s + 9-s − 6·11-s + 2·12-s + 8·17-s + 6·19-s − 12·21-s + 2·23-s − 2·27-s − 6·28-s + 2·29-s − 12·33-s + 36-s + 12·37-s + 36·41-s + 2·43-s − 6·44-s + 8·49-s + 16·51-s + 12·53-s + 12·57-s + 8·61-s − 6·63-s − 64-s + 42·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 2.26·7-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 1.94·17-s + 1.37·19-s − 2.61·21-s + 0.417·23-s − 0.384·27-s − 1.13·28-s + 0.371·29-s − 2.08·33-s + 1/6·36-s + 1.97·37-s + 5.62·41-s + 0.304·43-s − 0.904·44-s + 8/7·49-s + 2.24·51-s + 1.64·53-s + 1.58·57-s + 1.02·61-s − 0.755·63-s − 1/8·64-s + 5.13·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.474892911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.474892911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + 48 p T^{4} - 12 p^{2} T^{5} + 85 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T + 621 T^{2} - 6804 T^{3} + 51752 T^{4} - 6804 p T^{5} + 621 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 148 T^{3} - 1877 T^{4} + 148 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 42 T + 868 T^{2} - 11760 T^{3} + 113307 T^{4} - 11760 p T^{5} + 868 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 14307 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 202 T^{2} + 1848 T^{3} + 20067 T^{4} + 1848 p T^{5} + 202 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.67106704244770423842933119036, −6.34677380575521993574943930300, −5.93878941861164290816487413109, −5.76908142816943838228611766825, −5.75631243232626555413781224685, −5.67480403986000409170212217760, −5.32376443054217428520377202082, −5.20613980985200007037442651216, −4.93636391552560156415034608678, −4.24680890881607784523993358335, −4.24515032070989903854532850534, −4.24027703156403770087322553962, −3.82833653798591209144892445658, −3.48812757672656145818868770554, −3.37421997135436850283389383238, −2.97652263717091875013959214311, −2.95367496971954104060929628859, −2.67676563492584604856053156722, −2.46784770139233999295301256328, −2.38255743788492795765107849161, −2.19564666206182251916709746146, −1.24489510305999793323676883842, −1.09577675853303949248725458752, −0.876542969013062183631029622083, −0.37364629654869154881557387663,
0.37364629654869154881557387663, 0.876542969013062183631029622083, 1.09577675853303949248725458752, 1.24489510305999793323676883842, 2.19564666206182251916709746146, 2.38255743788492795765107849161, 2.46784770139233999295301256328, 2.67676563492584604856053156722, 2.95367496971954104060929628859, 2.97652263717091875013959214311, 3.37421997135436850283389383238, 3.48812757672656145818868770554, 3.82833653798591209144892445658, 4.24027703156403770087322553962, 4.24515032070989903854532850534, 4.24680890881607784523993358335, 4.93636391552560156415034608678, 5.20613980985200007037442651216, 5.32376443054217428520377202082, 5.67480403986000409170212217760, 5.75631243232626555413781224685, 5.76908142816943838228611766825, 5.93878941861164290816487413109, 6.34677380575521993574943930300, 6.67106704244770423842933119036
